Divide using synthetic division $$ ( -x^{3}-3x^{2}+11x+5 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&-1&-3&11&5\\& & 5& -10& \color{black}{-5} \\ \hline &\color{blue}{-1}&\color{blue}{2}&\color{blue}{1}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -x^{3}-3x^{2}+11x+5 }{ x+5 } = \color{blue}{-x^{2}+2x+1} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&-1&-3&11&5\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ -1 }&-3&11&5\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&-1&-3&11&5\\& & \color{blue}{5} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 5 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}-5&-1&\color{orangered}{ -3 }&11&5\\& & \color{orangered}{5} & & \\ \hline &-1&\color{orangered}{2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 2 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&-1&-3&11&5\\& & 5& \color{blue}{-10} & \\ \hline &-1&\color{blue}{2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}-5&-1&-3&\color{orangered}{ 11 }&5\\& & 5& \color{orangered}{-10} & \\ \hline &-1&2&\color{orangered}{1}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&-1&-3&11&5\\& & 5& -10& \color{blue}{-5} \\ \hline &-1&2&\color{blue}{1}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-5&-1&-3&11&\color{orangered}{ 5 }\\& & 5& -10& \color{orangered}{-5} \\ \hline &\color{blue}{-1}&\color{blue}{2}&\color{blue}{1}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}+2x+1 } $ with a remainder of $ \color{red}{ 0 } $.